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Mathematisches Forschungsinstitut Oberwolfach Representation Mathematisches Forschungsinstitut Oberwolfach Report No. 03/2014 DOI: 10.4171/OWR/2014/03 Representation Theory and Analysis of Reductive Groups: Spherical Spaces and Hecke Algebras Organised by Bernhard Kr¨otz, Paderborn Eric M. Opdam, Amsterdam Henrik Schlichtkrull, Copenhagen Peter Trapa, Salt Lake City 19 January – 25 January 2014 Abstract. The workshop gave an overview of current research in the rep- resentation theory and analysis of reductive Lie groups and its relation to spherical varieties and Hecke algebras. The participants and the speakers represented an international blend of senior researchers and young scientists at the start of their career. Some particular topics covered in the 30 talks related to structure theory of spherical varieties, p-adic symmetric spaces, symmetry breaking operators, automorphic forms, and local Langlands cor- respondence. Mathematics Subject Classification (2010): 22Exx, 20Cxx. Introduction by the Organisers The international conference Representation Theory and Analysis of Reductive Groups: Spherical Spaces and Hecke Algebras, organized by Bernhard Kr¨otz (Pader- born), Eric M. Opdam (Amsterdam), Henrik Schlichtkrull (Copenhagen), and Pe- ter Trapa (Salt Lake City) was held January 19th – January 25th, 2014. This conference brought together scientists from the separate, yet intimately related, fields of harmonic analysis of real Lie groups and representation theory of Hecke algebras. The meeting was attended by 50 participants, and a total of 30 lectures of length between 1/2 hour and 50 minutes were given. The participants and the speakers represented an international blend of senior researchers and young scientists at the start of their career. The meeting belongs to a long tradition of workshops around the theme of Harmonische Analysis und Darstellungstheorie 146 Oberwolfach Report 03/2014 topologischer Gruppen, but for most of the participants this was their first visit to Oberwolfach (and for some, even to Europe). The workshop opened with a lecture of Friedrich Knop giving an overview of the structure of spherical spaces, first in the case of complex reductive group actions and then in the much more recently established real case. The theory in the real case is foundational for approaching a host of geometric and analytic questions with very far reaching applications. For example, the space of unramified Langlands parameters for a reductive p-adic group in a great number of cases (and, optimistically, all cases) arises from the structure theory of certain spherical spaces of Langlands parameters for real reductive groups. The theory covered in Knop’s lectures should, therefore, provide insight into the intricate relationship between the representation theory of real and p-adic groups, something envisioned in rough form in the original work of Harish-Chandra and Langlands on the subject. The talk of Friedrich Knop was followed by an equally impressing talk by Patrick Delorme covering a new and interesting approach to the harmonic analysis on p- adic reductive symmetric spaces. Previous results by Sakellaridis and Venkatesh were put in a geometric perspective which opens up a wide possibility for gener- alization. All the selected speakers gave interesting talks of high quality in which recent research results were presented, and they were followed by vivid discussion among the participants. Also the talks by the young participants were noteworthy for their presenta- tion of new methods in the field. For example it was interesting to see in the talk by Benjamin Harris how microlocal techniques can provide some very basic information on representation of real reductive Lie groups. The relationship between representation theory of real and p-adic groups was again on display in the talk of Kei Yuen Chan, one of the workshop’s graduate student participants. His talk sought to transport ideas for the construction of discrete series representations of real reductive groups to the p-adic case via the theory of affine Hecke algebras. Chan’s talk sparked a suggestion from a senior workshop participant, namely to use deformations in the affine Hecke algebra setting (a tool unavailable in the real case), which Chan is now implementing to great effect. The workshop provided an ideal setting to encourage this kind of interaction between junior and senior participants. In the talk of Allen Moy it was shown that the sum of the depth zero Bernstein projectors for p-adic SL(2) is supported on the set of topologically unipotent el- ements (joint work with Howe). This beautiful result raises interesting questions for higher depth and more general groups. Moy’s talk was dedicated to the mem- ory of Paul Sally, who passed away a few weeks before the meeting. The early representation theory for p-adic SL(2;R) was to a large extend developed by Paul, whose passion for the topic is legendary. Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1049268, “US Junior Oberwolfach Fellows”. Representation Theory and Analysis of Reductive Groups 147 Workshop: Representation Theory and Analysis of Reductive Groups: Spherical Spaces and Hecke Algebras Table of Contents Friedrich Knop (joint with B. Kr¨otz, E. Sayag, H. Schlichtkrull) The local structure theorem ....................................... 149 Patrick Delorme Neighborhoods at infinity and the Plancherel formula for a reductive p-adic symmetric space ........................................... 151 Jan M¨ollers (joint with Yoshiki Oshima, Bent Ørsted) Model intertwining operators for symmetric pairs .................... 151 Job J. Kuit (joint with Erik P. van den Ban, Henrik Schlichtkrull) Cusp forms for reductive symmetric spaces ......................... 154 Vladimir L. Popov Quotients by conjugation action, cross-sections, singularities, and representation rings ............................................. 156 Birgit Speh Symmetry breaking for real rank one orthogonal groups and the Gross Prasad conjectures. Examples and Conjectures ...................... 159 Kayue Daniel Wong Regular Functions of Classical Nilpotent Orbits and Quantization ...... 160 Arnab Mitra On irreducible representations of GL2n(F ) with a symplectic period .... 161 Werner M¨uller (joint with Jonathan Pfaff) Approximation of L2-invariants of locally symmetric spaces ........... 163 Andre Reznikov (joint with Joseph Bernstein) Adelic action on periods of automorphic representations and special values of L-functions ............................................ 166 Benjamin Schwarz Hilbert series of Cauchy–Riemann filtrations ........................ 169 Raul Gomez (joint with Wee Teck Gan, Chen-Bo Zhu) On a conjecture of Sakellaridis-Venkatesh on the unitary spectrum of a spherical variety ................................................. 172 Dmitry Gourevitch (joint with Avraham Aizenbud, Eitan Sayag) Distributions on p-adic groups, finite under the action of the Bernstein center ................................................... ....... 172 148 Oberwolfach Report 03/2014 Toshiyuki Kobayashi Analysis on real spherical manifolds and their applications to Shintani functions and symmetry breaking operators ................. 176 Kei Yuen Chan Tempered modules with non-zero Dirac cohomology for graded affine Hecke algebras .................................................. 180 Benjamin Harris (joint with Hongyu He, Gestur Olafsson) Towards Asymptotic Plancherel Formulas for Reductive Homogeneous Spaces ................................................... ...... 181 Bogdan Ion (joint with S. Sahi) Automorphisms of double affine Hecke algebras ...................... 183 Dmitry Timashev On quotients of spherical varieties by unipotent subgroups ............ 184 Gang Liu (joint with Avraham Aizenbud, Dmitry Gourevitch, Bernhard Kr¨otz) Lie algebra homology of Schwartz spaces and comparison theorems ..... 187 Dmitry Akhiezer Equivariant real structures on spherical varieties .................... 188 Dipendra Prasad Ext analogues of branching laws ................................... 190 Shaun Stevens (joint with Corinne Blondel, Guy Henniart) On L-packets for p-adic symplectic groups .......................... 193 Allen Moy A computation with Bernstein projectors of depth 0 for p-adic SL(2) ... 195 Aaron Wood A Hecke algebra correspondence for the metaplectic group over Q2 ..... 195 Kyo Nishiyama Conormal variety over a double flag variety and exotic nilpotent cone .. 197 Chen-bo Zhu (joint with Binyong Sun) Conservation relations for local theta correspondence ................. 198 Tobias Finis (joint with Erez Lapid, Werner M¨uller) Limit multiplicities for congruence subgroups of arithmetic lattices ..... 202 Maarten Solleveld (joint with Anne-Marie Aubert, Paul Baum, Roger Plymen) The local Langlands correspondence for inner forms of SLn ........... 204 Jeffrey Adams Galois Cohomology of Real Groups ................................ 205 Erik P. van den Ban (joint with Dana Balibanu) Convexity theorems for semisimple symmetric spaces ................. 206 Representation Theory and Analysis of Reductive Groups 149 Abstracts The local structure theorem Friedrich Knop (joint work with B. Kr¨otz, E. Sayag, H. Schlichtkrull) Let G be a connected reductive group defined over C. Definition 1. The action of G on a variety X is elementary if the derived subgroup (G, G) acts trivially on X. In other words, the G-action on X factors through a torus action. In [1], Brion,
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